A topology on E-theory

For separable $C^{\ast }$-algebras $A$ and $B$, we define a topology on the set $[[A,B]]$ consisting of homotopy classes of asymptotic morphisms from $A$ to $B$. This gives an enrichment of the Connes–Higson asymptotic category over topological spaces. We show that the Hausdorffization of this category is equivalent to the shape category of Dadarlat. As an application, we obtain a topology on the $E$-theory group $E(A,B)$ with properties analogous to those of the topology on $KK(A,B)$. The Hausdorffized $E$-theory group $EL(A,B)=E(A,B)\mathrm{/}\overline{\{0\}}$ is also introduced and studied. We obtain a continuity result for the functor $EL(\cdot ,B)$, which implies a new continuity result for the functor $KL(\cdot ,B)$.